You allude to the best argument for these things, something like: "people are born into the world with a set of advantages and disadvantages they neither earned nor deserve, so we should do our best to even the playing field".
But you missed the mark in terms of identifying the best argument from the other side. "Because the market will solve it and those that work hard enough will make it, so it's fair" is not the argument. That's a just world hypothesis. The good argument is: high taxes and large social programs distort economic incentives; if you "tax the rich at 70%," people will behave differently, and your society will be less wealthy as a result. Which would you rather have? A wealthy society with inequality or a poor society without inequality? Anyway, the fundamental point here is that all human progress is agonistic, and "evening the playing field" will result in less progress, not more.
We're trying to walk the line between these two perspectives. We have "free" social programs but wealth still buys all kinds of advantages. I think we're doing a fairly good job, but it's imperative that we take both sides of this argument seriously. There's no reason to think we can have our cake and eat it.